Weak Gravitational Lensing of High-redshift 21 Cm Power Spectra

We describe the effects of weak gravitational lensing by cosmological large scale structure on the diffuse emission of 21 centimeter radiation from neutral hydrogen at high redshifts during the era of reionization. The ability to observe radial information through the frequency, and thus threedimensional regions of the background radiation at different redshifts, suggests that 21 cm studies may provide a useful context for studying weak lensing effects. We focus on the gravitational lensing effects on both the angular power spectra and the intrinsic, three-dimensional power spectra. We present a new approach for calculating the weak lensing signature based on integrating differential Fourier-space shells of the deflection field and approximating the magnification matrix. This reduces the problem of calculating the effect and higher order corrections to solving coupled systems of differential equations. This method is applied to reionization models of the 21 cm background spectra up to small angular scales over a range in redshift. The effect on the angular power spectrum is typically less than one percent on small angular scales, and very small on scales corresponding to the feature imprinted by reionization bubbles, due to the near-scale invariance of the angular power spectrum of the 21 cm signal on these scales. We describe the expected effect of weak lensing on three-dimensional 21 cm power spectra, and show that lensing creates aspherical perturbations to the intrinsic power spectrum which depend on the polar angle of the wavevector. The effect on the three-dimensional power spectrum is less than a percent on scales k . 0.1 h Mpc, but can be & 1% for highly inclined modes for k & 1 h Mpc. The angular variation of the lensing effect on these scales is well described by a quartic polynomial in the cosine of the polar angle. The detection of the gravitational lensing effects on 21 cm power spectra will require very sensitive, high resolution observations by future low-frequency radio arrays. Subject headings: cosmology: theory — diffuse radiation — gravitational lensing